It’s not that mysterious though.
Anyone carrying a bladed weapon carried oil. (More on that in a sec) Oil is what you use to clean and condition steel, especially, since water will rust it.
Many people in the Middle Ages used scented oils for their skin and hair from noblemen to lowly serfs.
Oil was incredibly abundant and quite cheap. The TYPE of oil however does matter in this.
Sheep oil (rendered from their fat) was very common and used for all manner of things from making soap to treating skin conditions. Rendered sheep fat has a very light texture and is a decent carrier oil without too pungent of a scent. Unfortunately it did rancid fast so it was common to add lots of herbs to it to help preserve it, especially rosemary, borage, marjoram and citron peels. This is how it became a common “perfume” oil used to scent hair skin or clothes. Nearly anyone would have had this handy somewhere.
Rendered pork oil was very common too and was most popular as a cooking oil.
Vegetable oil made from walnuts, almonds and flax seed was by far the most common non-animal oil. Nearly anybody had a bottle of almond or walnut oil in their pantry or on their person. These were by far the most popular oils used for conditioning steel, with walnut oil preferred because its tannins also gave armor a patina that kept it better. Only the absurdly wealthy ever wore polished armor. Everyone else blackened it to make it keep better. Walnut oil is good at doing that.
Walnut oil also works well as a lubricant. People back then DID use sexual lube by the way. No prostitute would be caught dead without it. Their favorite types were walnut and olive oil, though almond oil might be used in a pinch. They also used watered down acacia gum in southern Europe, which was sticky but slick and easy to re-wet.
Olive oil though was THE oil in Europe. It was expensive, comparatively, but obviously people considered it well worth its cost because it was found everywhere south of the Seine and frequently seen in even minor lordly houses or knights quarters much farther north. Considering quite a few people of the time thought it had aphrodisiac qualities when applied as certain way (likely because raw olive oil has a warming effect) I think you can imagine the most common reason it was sought after by men in particular.
Olive oil was also used in medicine and just about any church had some floating around somewhere because it’s conveniently good at treating minor infections and is wonderful for toothaches.
So the mysterious vial of oil isn’t at all mysterious and even if he were carrying it around with the sole intention of using it for sex, that wouldn’t actually be that strange either.
